Mathematical model
A non-isothermal dynamic model is developed based on the following
assumptions15-17:
- Axial dispersion-plug flow with radial changes in temperature and
concentration being negligible.
- Given the pressure of the process under study, Peng-Robinson equation
of state is utilized to describe gas parameters. Liquid properties are
not discussed here.
- The linear driving force (LDF) governs the rate of adsorption and a
lumped kinetic model is used for diffusion and adsorption into the
particles.
- Bed wall is fully insulated.
- Pressure drop along bed obeys Ergun equation.
- Heat capacities are constant.
- Constant porosity of the bed along the column.
Mass balance of component i in the gas phase yields to:
Eq. (1)
Axial dispersion is determined as follows:
Eq. (2)
Effective diffusion is calculated as follows:
Eq. (3)
Molecular diffusion is computed as follows:
Eq. (4)
Eq. (5)
Overall mass transfer balance is as follows:
Eq. (6)
Solid phase micro-pore mass balance is described by LDF:
Eq. (7)
Overall mass transfer coefficient and its corresponding resistance are
calculated as follows:
Eq. (8)
Film mass transfer coefficient is calculated as follows:
Eq. (9)
Pore diffusivity of component i is:
Eq. (10)
The Knudsen diffusion coefficient is:
Eq. (11)
The tortuosity factor is computed as follows:
Eq. (12)
The pressure drop equation is:
Eq. (13)
Bulk phase (gas) energy balance is:
Eq. (14)
Solid phase (gas) energy balance is:
Eq. (15)
Energy balance for wall thickness is calculated by:
Eq. (16)
The effective axial bed thermal conductivity is estimated as follows:
Eq. (17)
The extended Langmuir adsorption isotherm is used as follows:
Eq. (18)
Isotherm parameters of diethyl ether and ethanol are extracted from
literature. 12,15
All these governing equations are detailed elsewhere.16-17 The governing equations are simultaneously
solved using the finite difference method. To solve the equations,
initial condition is assumed as:
t=0 & 0<z<L Ci=0 T=293.15 K Eq.
(19)
Boundary conditions for mass and energy balance equations are as
follows:
@z=0 Ci=Cinlet,i and
T=Tinlet Eq. (20)
@ z=L and Eq. (21)